Galaxy Shape Measurement for
Radio Weak Lensing
Marzia Rivi
[email protected]
Department of Physics & Astronomy
University College London
Statistical Challenges in 21st Century Cosmology - Chania, Crete
May 27th, 2016
Why Radio Weak Lensing?
New generation of radio telescopes such as SKA will reach sufficient
sensitivity to provide large number density of faint galaxies.
SKA1: „ 3 gal/arcmin2 , SKA2: „ 10 gal/arcmin2
Higher redshift source distribution (beyond
LSST and Euclid).
Spectroscopic HI 21 cm line redshifts
Well-known and deterministic
knowledge of PSF solves one of the
biggest systematic errors.
Image credit: M. Brown et al. (2015)
Other measurements allows to mitigate intrinsic alignments:
polarization (Brown & Battye 2011, Whittaker et al 2015),
HI rotational velocity (e.g. Morales 2006)
Cross-correlation of shear estimators of optical and radio surveys drops
out wavelength dependent systematics.
Some Relevant Surveys Requirements
in terms of multiplicative (m) and additive (c) biases:
γobs ´ γtrue “ mγtrue ` c
or compressed into a single value Q “ 10´4 {pxm2 σ2γ ` c 2 yq.
Amara & Refregier (2008), Brown et al. (2015)
Galaxy Shape Measurement in the Radio Band
State-of-the-art optical lensing measurement fits model surface brightness
distributions to star-forming galaxies.
At „ 1GHz faint galaxies flux densities should be dominated by
synchrotron radiation emitted by the interstellar medium in the disk alone.
Galaxy models:
Shapelets
Sérsic brightness profiles, typically the disk component is modelled
by index 1 (exponential disk).
Fitting domain: image or visibility (Fourier)?
Galaxy Model: Shapelets
Shape decomposition through an 2D orthonormal
Gauss-Hermite basis functions (Refregier 2003):
I pxq “
ÿ
fn Bn px; βq
n
2
Bn px; βq “
Hn1 pβ´1 x1 qHn2 pβ´1 x2 qe
´ 2|xβ| 2
1
βr2n1 `n2 πn1 !n2 !s 2
Hm pξq is the Hermite polynomial of order m, n “ pn1 , n2 q, expansion truncated at n1 ` n2 ď Nmax
Fast convergence if β and x “ 0 are close to the size and location of the source.
ê “
Q11 ´ Q22 ` 2iQ12
1
2 2
Q11 ` Q22 ` 2pQ11 Q22 ´ Q12
q
,
Qij quadrupole moments
Invariant under Fourier Transform: FpBn px; βqq “ i pn1 `n2 q Bn pk; β´1 q.
Linear model: analytical solution of normal equations
Optical: model bias (Melchior 2010)
Galaxy Model: Exponential Disk
Sérsic surface brightness profile of index 1:
I pr q “ I0 e´r {α
Made it elliptical and rotated according to a linear transformation
dependent on the ellipticity parameters:
„
Apxq “
1 ´ e1
´e2
´e2
1 ` e1
„
x1
x2

Analytical Fourier Transform of the model:
2πα2 I0
p1 ` 4π2 α2 k 2 q3{2
1
FpI ˝ Aqpkq “
FpI pr qqpA´T kq
det A
FpI pr qqpk q “
Optical: good performance in the GREAT Challenges of methods using
Sérsic models, they reduce model bias observed with shapelets.
Measurement in the Image Domain
complex raw visibilities
ż ż
V pu , v ; ν, t q „
Iν pl , m; t qe´2πi pul `vmq dldm
Ó calibration, gridding, F.T.
dirty image
ż ż
ID “
S pu , v qV pu , v qe2πi pul `vmq dudv “ I ˚ PSF
Ó iterative PSF deconvolution
clean image
optical techniques can be used to fit a galaxy at a time
non linear imaging procedure may introduce systematic spurious
shear signal
the noise is highly correlated in the image
Measurement in the image domain
Figure: Ian Harrison
Measurement in the Visibility Domain
uv coverage: raw visibilities are sampled at
discrete locations in the uv plane (orthogonal to
the antennas pointing direction)
sampling function: S pu , v q “
ř
i
δpu ´ ui , v ´ vi q
Many visibilities ñ Gridding the uv coverage
SKA1: 197 dishes, SKA2: „ 2000 dishes
SKA1: 19,306 baselines per frequency channel per time sampling!
Sources are not localised
Further parameters to fit
Positions from the clean image
Flux from all sources within a pointing is mixed together
ñ Joint fitting, ...
Measurements from VLA FIRST Survey
About 10, 000 deg2 at 1.4 GHz, 14 3-MHz frequency channels, 3-min snapshots
detection threshold 1 mJy, „ 30 resolved sources deg´2 .
Shapelets fitting:
Binning of the visibilities
Shapelet centroid, β and Nmax set from the source position and FWHMs in the FIRST
catalog.
For each pointing p„ 50, 000q, sources are fitted simultaneously.
Shear estimator for each galaxy from polar shapelet coefficients, linearly related to
Cartesian coefficients (Refregier & Bacon 2003):
?
γ̂s “
2 f̂21,2
xf̂0,0 ´ f̂4,0 y
,
ws “ source S/N
γ̂ “
Σs ws2 γ̂s
.
Σs ws2
Corrections for systematics: removed artificial shear due to (i) sampling function (PSF),
(ii) non-coplanarity, (iii) bandwidth and time smearing, (iv) primary beam attenuation.
3.6σ detection of cosmic shear.
Chang et al. (2004)
RadioLensfit - single source
Lensfit (Miller et al. 2013) adaptation to visibility domain
Chi-square fitting:
Exponential disk model visibilities, 6 free parameters
LpS , x, α, eq 9 e´
χ2
2
,
χ2 “ pD ´ MS q: C´1 pD ´ MS q
Bayesian marginalisation over S , x, α ñ Lpeq
analytical over flux S by adopting uniform prior
straightforward over position x by adopting uniform prior (exponential
integral computation)
numerical over scalelength α by adopting lognormal prior dependent
on source flux (derived from VLA 20cm survey in the SWIRE field)
Likelihood sampling: ML + adaptive grid around the maximum
ê = Likelihood
mean point
a
σ2e “ detpΣq, where Σ is the likelihood covariance matrix
Rivi, Miller et al. (2016), arXiv:1603.04784
RadioLensfit - shear bias
SKA1-MID 8 hour track observation, ∆t “ 60 s,
bandwidth: 950 - 1190 MHz, 12 channels.
Reduced shear measurements from visibilities of individual
galaxies at the phase centre.
1% accuracy: 104 galaxies (flux range 10 ´ 200µJy).
Input shear : g “ 0.04 with 8 different orientations and g “ 0.
Fitting of Many Sources in the Primary Beam
SKA1: „ 104 sources!
Is a joint fitting for high dimension problems feasible?
Shapelets: huge matrix size for many sources
MultiNest: single source model, search for peaks in the multimodal
posterior by computing the Bayesian evidence.
+ Can be also used to localise sources
- Many peaks, very slow even with a small number of parameters
Hamiltonian Monte Carlo: deterministic proposal for parameters using
Hamiltonian dynamics. (Work in progress using GPUs, collaboration with S. Balan, M.
Lochner, F.B. Abdalla)
+ Reasonable efficiency even for high dimensional problems, (e.g.
Taylor et al. 2008, fit of 105 parameters for CMB power spectrum).
- Posterior gradient required.
All computationally very expensive for SKA!
Fitting of Many Sources in the Primary Beam
Can we extract and fit single or subsamples of sources?
Extraction from the dirty map:
1. extract postage stamp
2. inverse Fourier Transform
Facetting:
1. visibilities phase shifting to source
position
2. use a coarse grid
Apodisation effects and contamination from
sources close to the region of interest must be
assessed.
collaboration with L. Miller
radioGreat Challenge
Forthcoming challenge to benchmark current and new methods
for shape measurement from radio data.
High quality simulated data sets of observations of fields of radio
galaxies for participants to use to attempt to blindly measure the
applied shear.
Both image and visibility-plane versions of the data sets
available.
http://radiogreat.jb.man.ac.uk/
Conclusions
New generation of radio telescope such as SKA will allow weak lensing
in the radio wavelenghts comparable to optical.
Radio data originate in the Fourier space and standard imaging
technique (CLEAN) is not accurate enough to allow image-based methods
for shape measurement.
Methods in the visibility domain are so far the only ones to successfully
detect radio weak lensing but are computationally very challenging for
SKA datasets.
Bayesian methods can use more accurate galaxy models and reduce
shear bias, but still open discussion for the fitting of many sources in the
visibility domain.
Joint analysis in the image/visibility domains can be required for SKA.